
Notations and terminology for ODE's and systems of ODE's;
reduction of higher order ODE's to 1st order systems of ODE's; the
fundamental existence and uniqueness thm. for ODE's (Lipschitz condition) 

Introduction of Euler's method, order of Euler's method,
one step methods (introduction, definition, consistency, local truncation
error) 

Explicit RungeKutta (ERK) methods (introduction of the
method in the general case, notations in the general case, derivation of ERK
of second order); RungeKutta method of fourth order 

Examples of implicit methods: trapezoidal rule, midpoint
rule, the theta method, and the implicit Euler's method; computation of
orders for these methods 

Convergence of onestep methods (the general case; see
also convergence for Euler's method, etc) 

Asymptotic expansions for the global discretization error
for one step methods, and applications to error estimate 

Practical implementation of one step methods 

Linear Multistep methods: examples, derivation using the
Lagrange interpolation polynomial 

Linear multistep methods: definition and computations of
the local truncation error, order of the method, consistency 

Implicit and explicit linear multistep methods,
predictorcorrector methods 

Examples of consistent multistep methods which diverge 

Linear difference equations: stability (root) condition,
general solution 

Convergence Thm. for linear multistep methods 

Order and consistency for linear multistep methods 

Adaptive methods for onestep and multistep methods,
error control, Milne device, extrapolation 

Stiff differential equations, stability and intervals
(regions) of absolute stability, Astable methods, BDF methods 

Numerical methods and stability for systems of ODE's 

Finite difference methods for linear BVP 

Functional (fixed point) iteration and Newton's iteration
for solving systems of ODE's using an implicit method 